Quantum Vacuum Energy Calculations: From Graphs to ...lkaplan/graphs_polyhedra.pdfQuantum Vacuum...
Transcript of Quantum Vacuum Energy Calculations: From Graphs to ...lkaplan/graphs_polyhedra.pdfQuantum Vacuum...
Quantum Vacuum Energy Calculations: From Graphs toPolyhedra and Beyond
Lev Kaplan
Tulane University
March 30, 2011
Mathematical Physics and Harmonic Analysis SeminarTexas A&M University
Lev Kaplan (Tulane University) Quantum Vacuum Energy March 30, 2011 1 / 49
Texas-Oklahoma-Louisiana Quantum VacuumCollaboration:
Ricardo Estrada (LSU)
Stephen A. Fulling (Texas A&M)
Lev Kaplan (Tulane)
Klaus Kirsten (Baylor)
Kimball A. Milton (U. Oklahoma)
Lev Kaplan (Tulane University) Quantum Vacuum Energy March 30, 2011 2 / 49
Outline
1 IntroductionHistory and BackgroundMathematical PreliminariesBroad Objectives
2 ExamplesEx. 1: Quantum GraphsEx. 2: Rectangles, Pistons, and PistolsEx. 3: Triangles, Triangular Cylinders, Prisms, and Tetrahedra
3 OutlookRegular PolygonsChaotic CavitiesOutside of a Box
4 Conclusions
Lev Kaplan (Tulane University) Quantum Vacuum Energy March 30, 2011 3 / 49
Introduction History and Background
History and Background
Any quantum system has nonzero ground state energy, e.g. ~ω/2 forharmonic oscillator with frequency ω
In particular, vacuum state of a quantum field has nonzero energy,which is a function of boundary conditions
1948: Casimir-Polder force predicted between uncharged conductingplates
F/A = −π2~c/240a4
Measurement in 1958consistent with theory
Lev Kaplan (Tulane University) Quantum Vacuum Energy March 30, 2011 4 / 49
Introduction History and Background
History and Background
1997: More accurate experiment using plane and sphere
2002: Finally, original Casimir parallel plate experiment performedwith 15% precision
Lev Kaplan (Tulane University) Quantum Vacuum Energy March 30, 2011 5 / 49
Introduction History and Background
Vacuum energy beyond parallel plates:
Relation to van der Waals force in chemistry
1970s: QFT in curved spacetime
Chiral bag model of the nucleon
Dark energy in cosmology
Λ ∼ 10−26kg/m3 ∼ 10−120c5/~G 2
Why so small?Why not zero?Why comparable to present-day mass density of universe?
Lev Kaplan (Tulane University) Quantum Vacuum Energy March 30, 2011 6 / 49
Introduction History and Background
Stabilizing extra dimensions in brane world models
Nanotechnology applications: MEMS and NEMS
Static friction caused by vacuumenergy is major obstacle to furtherminiaturization of devices such asgears, etc.
Lev Kaplan (Tulane University) Quantum Vacuum Energy March 30, 2011 7 / 49
Introduction Mathematical Preliminaries
Mathematical Preliminaries
Vacuum energy of scalar field in cavity
Work in units ~ = c = 1
Find all modes of field φ consistent with given boundary conditions,i.e. solve
−∇2φn = ω2nφn
Each mode n behaves as independent harmonic oscillator withfrequency ωn
Each mode has energy ωn(Nn + 12) Nn = 0, 1, 2, . . .
Total vacuum energy of the field is given formally by
Evac =1
2
∑n
ωn =1
2Tr√−∇2
Lev Kaplan (Tulane University) Quantum Vacuum Energy March 30, 2011 8 / 49
Introduction Mathematical Preliminaries
Mathematical Preliminaries
Evac = 12
∑n ωn = 1
2 Tr√−∇2 divergent, must regularize
Cylinder (Poisson) kernel: Tt(x , y) = 〈x |e−t√−∇2 |y〉
Let Et = −1
2
∂
∂tTr Tt =
1
2
∑n
ωne−ωnt
Can separate finite and divergent parts as t → 0 and get t−independentfinite answer for physical forces if divergent terms can be shown to cancel[renormalization]
Similarly regularized local energy densityEt(x) = T00(x) = −1
2∂∂tTt(x , x) = 1
2
∑n ωn|φ(x)|2e−ωnt
Other components of stress-energy tensor can be obtained similarly
Lev Kaplan (Tulane University) Quantum Vacuum Energy March 30, 2011 9 / 49
Introduction Mathematical Preliminaries
Real life is more complicated
In specific applications need to consider
vector fields (e.g. electromagnetic fields)
going beyond idealized boundary conditions
more physically realistic cutoffs (e.g. spatial dispersion)
Advantages of toy model
can address general conceptual issues independent of specificapplication
mathematical problem in spectral geometry: asymptotics of cylinderkernel, relation to zeta function, ...
Lev Kaplan (Tulane University) Quantum Vacuum Energy March 30, 2011 10 / 49
Introduction Broad Objectives
Objectives/Questions
Role of periodic and closed classical orbits
quantum-classical correspondence
Sign of Casimir force in general situations
Relation of local and global quantities
nonuniform convergence
Systematic understanding of boundary, curvature, and corner effects
Proper renormalization
under what circumstances do divergent terms cancel?can all ∞’s be absorbed into boundary properties?
Combining “inside” + “outside” contributions
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Examples Ex. 1: Quantum Graphs
Ex. 1: Quantum Graphs
S. A. Fulling, L.K., and J. H. Wilson, PRA (2007)G. Berkolaiko, J. M. Harrison, and J. H. Wilson, J Phys A (2009)J. H. Wilson, Texas A&M senior thesis
Lev Kaplan (Tulane University) Quantum Vacuum Energy March 30, 2011 12 / 49
Examples Ex. 1: Quantum Graphs
What is a quantum graph?
Set of line segments joined at vertices
Singular one-dimensional variety equipped with self-adjoint differentialoperator
Approximation for realistic physical wave systems
Chemistry: free electron theory of conjugated moleculesNanotechnology: quantum wire circuitsOptics: photonic crystals
Laboratory for investigating general questions about scattering,quantum chaos, spectral theory
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Examples Ex. 1: Quantum Graphs
What is a quantum graph?
H = −∇2 on each bond
Kirchhoff boundary conditions at each vertex
1 Continuity φj(0) = φα for all bonds j starting at vertex α2 Current conservation
∑j φ′j(0) = cαφα where sum is over all bonds j
starting at vertex α, and derivative is in outward direction
Neumann-like: cα = 0; Dirichlet: cα =∞
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Examples Ex. 1: Quantum Graphs
Direct calculation using spectrum
Two movable “pistons”
Focus on “a” region between pistons:Dirichlet (DD): ωn = nπ/a (n = 1 . . .∞)Neumann (NN): ωn = nπ/a (n = 0 . . .∞)
Tr Tt =∑∞
n=0,1 e−(nπ/a)t = a
πt ±12 + 1
12πta + O(t2)
⇒ Et = −1
2
∂ Tr Tt
∂t=
a
2πt2− π
24a+ O(t)
Lev Kaplan (Tulane University) Quantum Vacuum Energy March 30, 2011 15 / 49
Examples Ex. 1: Quantum Graphs
Vacuum Energy in simple QG
Et =a
2πt2− π
24a+ O(t)
First term divergent as t → 0
ETotalt = a+L1+L2
2πt2− π
24(a−1 + L1−1 + L2
−1) + O(t)
= const2πt2− π
24(a−1 + L1−1 + L2
−1) + O(t)
Divergent term is a-independent constant energy density⇒ force on piston is finite!
Lev Kaplan (Tulane University) Quantum Vacuum Energy March 30, 2011 16 / 49
Examples Ex. 1: Quantum Graphs
Vacuum Energy in simple QG
ETotal = const− π24(a−1 + L1
−1 + L2−1)
Can safely take L1, L2 →∞
ETotalt = const− π
24a
⇒ FDD = FNN = −∂E∂a
= − π
24a2(attractive)
If one piston is Dirichlet and the other Neumann,
⇒ FDN = +π
48a2(repulsive)
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Examples Ex. 1: Quantum Graphs
Periodic Orbit Perspective
Tr Tt =∫dx Tt(x , x)
Free cylinder kernel in 1D: T 0t (x , y) = t
π1
(x−y)2+t2
Then Tt(x , x) in problem with boundaries obtainable by method ofimages as sum over periodic and closed orbits:
Tt(x , x) = Re∑p
t
π
Ap
L2p + t2+ closed orbits
p = periodic orbit going through xLp = orbit lengthAp = product of scattering factors
Can obtain asymptotic t → 0 behavior term by term
Lev Kaplan (Tulane University) Quantum Vacuum Energy March 30, 2011 18 / 49
Examples Ex. 1: Quantum Graphs
Periodic Orbit Perspective
Taking trace and accounting for repetitions r
Tr Tt =∫dx Tt(x , x) = t
πat2
+ Re∑
p
∑∞r=1
tπ2Lp(Ap)r
(rLp)2+ O(t2)
Divergent (Weyl) term associated with zero-length orbit
For single line segment a, only one nonzero-length periodic orbitLp = 2a plus repetitions
FDD = FNN = − 14πa2
(+1 + 14 + 1
9 + · · · )FDN = − 1
4πa2(−1 + 1
4 −19 + · · · )
Periodic orbit sum converges
Sign of force can be read off from phase associated with shortest orbit(in general, boundaries + Maslov indices)
Lev Kaplan (Tulane University) Quantum Vacuum Energy March 30, 2011 19 / 49
Examples Ex. 1: Quantum Graphs
Vacuum Energy in Star Graphs
• N-like boundary at junction joining Bbonds
• N, D, or e iθj boundary at each piston
Each piston at distance aj from junction
Eigenvalues: solve∑B
j=1 tan(ωaj + θj) = 0
Exact solution for ωn when all aj equal
⇒ Then Evac = limt→0
[Et −
∑j aj
2πt2
]= lim
t→0
[1
2
∑n
ωne−ωnt −
∑j aj
2πt2
]
Convergence improved by Richardson extrapolation
Lev Kaplan (Tulane University) Quantum Vacuum Energy March 30, 2011 20 / 49
Examples Ex. 1: Quantum Graphs
Alternatively: use periodic orbit expansion
Contribution to vacuum energy from shortest orbit only (bouncing backand forth once in one bond):
Evac ≈ −1
2π
(2
B− 1
) B∑j=1
(±1)
aj
±1 for Neumann or Dirichlet pistons
Gives correct sign for Casimir forces at least for B > 3
repulsive for Neumann
attractive for Dirichlet
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Examples Ex. 1: Quantum Graphs
Comparison between shortest orbit approximation & exactanswer for equal bond case
Lev Kaplan (Tulane University) Quantum Vacuum Energy March 30, 2011 22 / 49
Examples Ex. 1: Quantum Graphs
Star Graphs: Periodic Orbits
Add up all repetitions of shortest orbits (Neumann):
E ≈ − 1
4π
∞∑r=1
1
r2
(2
B− 1
)r B∑j=1
1
aj≈ π
48
(1− 24 ln 2
π2B+ · · ·
) B∑j=1
1
aj
Compare with analytic result for B Neumann pistons with equal bondlengths:
E =π
48
(1− 3
B
)B
a
Shortest orbits give only leading contribution in 1/B expansion
⇒ need more orbits to obtain full answer for finite B
Lev Kaplan (Tulane University) Quantum Vacuum Energy March 30, 2011 23 / 49
Examples Ex. 1: Quantum Graphs
Star Graphs: Periodic Orbit Sum
B = 4 star graph with unequal bonds and all Neumann pistons
Sum over orbits Lp ≤ Lmax & compare with “exact” answer
0
0.1
0.2
5 10 15 20 25 30 35 40
Casim
ir En
ergy
Lmax
Star graph with B=4 and all Neumann pistons
Periodic orbit sumExact
0.01
0.1
1 10
Erro
r in
Casim
ir En
ergy
Lmax
Star graph with B=4 and all Neumann pistons
ErrorLmax
-1 power law
Error ∼ (Lmax)−1Lev Kaplan (Tulane University) Quantum Vacuum Energy March 30, 2011 24 / 49
Examples Ex. 1: Quantum Graphs
Star Graphs: Periodic Orbit Sum
B = 4 star graph with unequal bonds and arbitrary e iθ pistons
Sum over orbits Lp ≤ Lmax & compare with “exact” answer
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0.05
5 10 15 20 25 30 35 40
Casim
ir En
ergy
Lmax
Star graph with B=4 and random phase pistons
Periodic orbit sumExact
0.0001
0.001
0.01
1 10
Erro
r in
Casim
ir En
ergy
Lmax
Star graph with B=4 and random phase pistons
ErrorLmax
-3/2 power law
Error ∼ (Lmax)−3/2Lev Kaplan (Tulane University) Quantum Vacuum Energy March 30, 2011 25 / 49
Examples Ex. 2: Rectangles, Pistons, and Pistols
Ex. 2: Rectangles, Pistons, and Pistols
S. A. Fulling, L.K., K. Kirsten, Z. H. Liu, and K. A. Milton,J Phys A (2009); Z. H. Liu, Texas A&M Ph.D. thesis
Motivating Question: Naive renormalization (Lukosz 1971, ...) suggestsoutward force on sides of square or cubic box
Is this force real?
Lev Kaplan (Tulane University) Quantum Vacuum Energy March 30, 2011 26 / 49
Examples Ex. 2: Rectangles, Pistons, and Pistols
Ex. 2a: Rectangular cavity
No straightforward way to eval-uate Et = 1
2
∑n ωne
−ωnt di-rectly
Use classical path approach
Need all classical paths from x to x
Classify by number of bounces from a sides and number of bounces from bsides
Periodic paths: Even Even
Side paths: Even Odd or Odd Even
Corner paths: Odd Odd
Lev Kaplan (Tulane University) Quantum Vacuum Energy March 30, 2011 27 / 49
Examples Ex. 2: Rectangles, Pistons, and Pistols
Rectangle: periodic paths
Et,Periodic =ab
2πt3− ab
2π
∞∑k=1
(−1)η(2kb)2 − 2t2
[t2 + (2kb)2]5/2
− ab
2π
∞∑j=1
(−1)η(2ja)2 − 2t2
[t2 + (2ja)2]5/2
−ab
π
∞∑j=1
∞∑k=1
(−1)η(2ja)2 + (2kb)2 − 2t2
[t2 + (2ja)2 + (2kb)2]5/2
η = # of Dirichlet bouncesAssume all Neumann or all Dirichlet sides:
Et,Periodic =ab
2πt3− ζ(3)
16π
(a
b2+
b
a2
)− ab
8π
∞∑j=1
∞∑k=1
(a2j2 + b2k2
)−3/2+ O(t2)
Lev Kaplan (Tulane University) Quantum Vacuum Energy March 30, 2011 28 / 49
Examples Ex. 2: Rectangles, Pistons, and Pistols
Rectangle: add all paths
Et,Nonperiodic = ∓ 2a + 2b
8πt2± π
48
(1
a+
1
b
)+ O(t2)
Combining all terms:
Et =Area
2πt3∓ Perimeter
8πt2− ζ(3)
16π
(a
b2+
b
a2
)− ab
8π
∞∑j ,k=1
(a2j2 + b2k2
)−3/2 ± π
48
(1
a+
1
b
)+ O(t2)
Force on horizontal side:
F = −∂Et
∂a= divergent +
ζ(3)
16πb2− ζ(3)b
8πa3
+b
8π
∞∑j ,k=1
k2b2 − 2j2a2
(j2a2 + k2b2)5/2± π
48a2
Lev Kaplan (Tulane University) Quantum Vacuum Energy March 30, 2011 29 / 49
Examples Ex. 2: Rectangles, Pistons, and Pistols
Force on side of rectangle
F = divergent +ζ(3)
16πb2− ζ(3)b
8πa3+
b
8π
∞∑j ,k=1
k2b2 − 2j2a2
(j2a2 + k2b2)5/2± π
48a2
Naive “renormalization”: drop divergent terms and interprett−independent result as physical force on the side of box
a << b: attractive (like parallel plates)
Square box with Dirichlet sides: repulsive!
Problems:
Throwing away infinite terms
Ignoring outside of box
Lev Kaplan (Tulane University) Quantum Vacuum Energy March 30, 2011 30 / 49
Examples Ex. 2: Rectangles, Pistons, and Pistols
Ex. 2b: Rectangular piston
Add contributions from a× b rectangle and (L− a)× b rectangle
Divergent terms cancel (total area and perimeter are conserved)
One finite contribution from (L− a)× b rectangle survives as L→∞
Fpiston = 0 +ζ(3)
16πb2− ζ(3)b
8πa3
+b
8π
∞∑j ,k=1
k2b2 − 2j2a2
(j2a2 + k2b2)5/2± π
48a2− ζ(3)
16πb2
Lev Kaplan (Tulane University) Quantum Vacuum Energy March 30, 2011 31 / 49
Examples Ex. 2: Rectangles, Pistons, and Pistols
Ex. 2b: Rectangular piston
Finally (Cavalcanti 2004)
Fpiston = −ζ(3)b
8πa3+
b
8π
∞∑j ,k=1
k2b2 − 2j2a2
(j2a2 + k2b2)5/2± π
48a2
=π
b2
∞∑j ,k=1
k2K ′1
(2πjk
a
b
)[always attractive!]
= − ζ(3)b
8πa3+
π
48a2− ζ(3)
16πb2+πb
a3
∞∑j ,k=1
k2K0
(2πjk
b
a
)Decays exponentially for a� b; parallel plates for a� b
Lev Kaplan (Tulane University) Quantum Vacuum Energy March 30, 2011 32 / 49
Examples Ex. 2: Rectangles, Pistons, and Pistols
Ex. 2c: Casimir pistol
What would happen if exter-nal shaft is not present?
Here all dimensions� cutofft, except possibly c ∼ t
All Dirichlet boundaries
E =us
πt
∞∑k=1
1− 2k2u2
(1 + 4k2u2)5/2+
us
πt
∞∑j=1
1− 2j2s2
(1 + 4j2s2)5/2
+2us
πt
∞∑j=1
∞∑k=1
1− 2j2s2 − 2k2u2
(1 + 4j2s2 + 4k2u2)5/2
+s
2πt
∞∑j=1
−1 + 4j2s2
(1 + 4j2s2)2+
2r(l − s)
πt
∞∑k=1
1− 2k2r2
(1 + 4k2r2)5/2
Lev Kaplan (Tulane University) Quantum Vacuum Energy March 30, 2011 33 / 49
Examples Ex. 2: Rectangles, Pistons, and Pistols
Ex. 2c: Casimir pistol
Use scaled variablesc = rta = stb = utd = L− a = (l − s)t
E =us
πt
∞∑k=1
1− 2k2u2
(1 + 4k2u2)5/2+
us
πt
∞∑j=1
1− 2j2s2
(1 + 4j2s2)5/2
+2us
πt
∞∑j=1
∞∑k=1
1− 2j2s2 − 2k2u2
(1 + 4j2s2 + 4k2u2)5/2
+s
2πt
∞∑j=1
−1 + 4j2s2
(1 + 4j2s2)2+
2r(l − s)
πt
∞∑k=1
1− 2k2r2
(1 + 4k2r2)5/2
Lev Kaplan (Tulane University) Quantum Vacuum Energy March 30, 2011 34 / 49
Examples Ex. 2: Rectangles, Pistons, and Pistols
Ex. 2c: Casimir pistol
Horizontal force as func-tion of a:
narrow chamber a� b1/3c2/3: like parallel platesF ∼ −1/a2 (attractive)
longer chamber a� b1/3c2/3: gaps dominate:F is a−independent
c > 0.6t ⇒ attractivec < 0.6t ⇒ repulsive (believable???)
Lev Kaplan (Tulane University) Quantum Vacuum Energy March 30, 2011 35 / 49
Examples Ex. 3: Triangles, Triangular Cylinders, Prisms, and Tetrahedra
Example 3: Triangles, Triangular Cylinders, Prisms, andTetrahedra
[E. K. Abalo, K. A. Milton, and LK (PRD, 2010)]
Analytic expressions exist for spectrum of the following triangular cavitieswith Dirichlet or Neumann boundaries
Equilateral triangle (60◦ − 60◦ − 60◦)
Hemiequilateral triangle (30◦ − 60◦ − 90◦)
Isosceles right triangle (45◦ − 45◦ − 90◦)
For other triangles, Dirichlet spectrum may be evaluated numerically
Scaling method of Vergini et al. - code by Alex Barnett
Lev Kaplan (Tulane University) Quantum Vacuum Energy March 30, 2011 36 / 49
Examples Ex. 3: Triangles, Triangular Cylinders, Prisms, and Tetrahedra
Calculating vacuum energy of infinite cylinder
Given spectrum ωm,n for 2-dimensional cross-section,
Vacuum energy per unit length is
Evac =1
2limt→0
∑m,n
∫ ∞−∞
dk
2π
√k2 + ω2
m,ne−t√
k2+ω2m,n
=1
2limt→0
(− d
dt
)∫ ∞−∞
dk
2π
∑m,n
e−t√
k2+ω2m,n
Lev Kaplan (Tulane University) Quantum Vacuum Energy March 30, 2011 37 / 49
Examples Ex. 3: Triangles, Triangular Cylinders, Prisms, and Tetrahedra
E.g. Equilateral Triangular Cylinder
Spectrum:
ω2mnp =
2π2
3h2(m2 + n2 + p2) with m + n + p = 0
ω2m,n =
4π2
3h2(m2 + mn + n2)
Degeneracy factor of 1/6
Dirichlet: m 6= 0, n 6= 0, and m + n 6= 0
Neumann: m 6= 0 or n 6= 0
Lev Kaplan (Tulane University) Quantum Vacuum Energy March 30, 2011 38 / 49
Examples Ex. 3: Triangles, Triangular Cylinders, Prisms, and Tetrahedra
E.g. Equilateral Triangular Cylinder
Vacuum energy:
Evac =1
2limt→0
(− d
dt
)∫ ∞−∞
dk
2π
∑m,n
e−t√
k2+ω2m,n
may be evaluated using Poisson’s summation formula
Isolate Divergences:
EDivvac =
(√3 h2
2π2t4±√
3 h
4πt3+
1
6πt2
)=
(3A
2π2t4± P
8πt3+
C
48πt2
)where A = Area, P = Perimeter, C =
∑i
(παi− αi
π
)
Lev Kaplan (Tulane University) Quantum Vacuum Energy March 30, 2011 39 / 49
Examples Ex. 3: Triangles, Triangular Cylinders, Prisms, and Tetrahedra
E.g. Equilateral Triangular Cylinder
Finally, we have finite part:
E(D,N)vac =
1
144π2h2
±12πζ(3)− 10√
3
3ζ(4)− 16
√3
3
∞∑p,q=1
1 + (−1)p+q
(p2 + q2/3)2
Further simplifying:
E(D,N)vac =
1
96h2
[√3
9
[ψ′(2/3)− ψ′(1/3)
]± 8
πζ(3)
]
Evaluating:
E(D)vac =
0.017789
h2E(N)
vac = −0.045982
h2
Lev Kaplan (Tulane University) Quantum Vacuum Energy March 30, 2011 40 / 49
Examples Ex. 3: Triangles, Triangular Cylinders, Prisms, and Tetrahedra
Results for Dirichlet Triangular Cylinders
Plotting the scaled (dimensionless) vacuum energy εA vs. dimensionlessgeometric parameter A/P2, we observe “universal” behavior:
0.02 0.04 0.06 0.08
A
P2
-6
-4
-2
2
lnHE AL
Lev Kaplan (Tulane University) Quantum Vacuum Energy March 30, 2011 41 / 49
Examples Ex. 3: Triangles, Triangular Cylinders, Prisms, and Tetrahedra
Results for Dirichlet Right Triangles
0.02 0.04 0.06 0.08
A
P2
-3
-2
-1
1
2
logK A EO
Numerical vacuum energy for Dirichlet right triangles plotted alongwith exact results for the three special triangles, and the square
Numerical and exact results agree within 1% where exact results exist
Curve is proximity force approximation for acute angle of righttriangle → 0 (including contribution from two long sides only)
Lev Kaplan (Tulane University) Quantum Vacuum Energy March 30, 2011 42 / 49
Examples Ex. 3: Triangles, Triangular Cylinders, Prisms, and Tetrahedra
Analogous Calculations for three special Tetrahedra
Lev Kaplan (Tulane University) Quantum Vacuum Energy March 30, 2011 43 / 49
Examples Ex. 3: Triangles, Triangular Cylinders, Prisms, and Tetrahedra
Use General Poisson resummation formula:
∞∑m1,...,mn=−∞
e−τ√
(m+a)j Ajk (m+a)k
=2nπ(n−1)/2 Γ((n + 1)/2)
|det (A)|1/2∞∑
m1,...,mn=−∞
τ e−2πi mj aj
(τ2 + 4π2kj kj)(n+1)/2
Lev Kaplan (Tulane University) Quantum Vacuum Energy March 30, 2011 44 / 49
Examples Ex. 3: Triangles, Triangular Cylinders, Prisms, and Tetrahedra
E.g. for “Large Tetrahedron”
Spectrum:
ω2kmn =
π2
4a2[3(k2 + m2 + n2)− 2(km + kn + mn)
]Dirichlet boundaries: k > m > n > 0
Vacuum energy:
Divergent terms involving volume, perimeter, corners, appear as before
Finite part is
E (D)vac =
1
a
{− 1
96π2[Z3(2; 1, 1, 1) + Z3b(2; 1, 1, 1)
]+
1
8πζ(3/2)L−8(3/2)
+1
16πZ2b(3/2; 2, 1)− π
96− π√
3
72
}= −0.0468804266
aLev Kaplan (Tulane University) Quantum Vacuum Energy March 30, 2011 45 / 49
Outlook Regular Polygons
Outlook: Regular Polygons
Exact results for triangle and square only
Can we interpolate between these and the circle?What happens when number of sides � 1?
Work in progress!
Lev Kaplan (Tulane University) Quantum Vacuum Energy March 30, 2011 46 / 49
Outlook Chaotic Cavities
Outlook: Chaotic cavities
Lev Kaplan (Tulane University) Quantum Vacuum Energy March 30, 2011 47 / 49
Outlook Outside of a Box
Outlook: Outside of a Box
Consider concentric polygons of size a and b
Efinitet =
1
a
∞∑p,q=0
Cp,q
(ab
)p ( ta
)q=
(C0,0
a+
C1,0
b+
C2,0a
b2+ · · ·
)+ t
(C0,1
a2+
C1,1
ab+
C2,1
b2+ · · ·
)+ O(t2)
Take t � a� b and extract C0,0
Work in progress!
Lev Kaplan (Tulane University) Quantum Vacuum Energy March 30, 2011 48 / 49
Conclusions
Conclusions
Careful regularization and renormalization (including inside and outsidecontributions) needed to obtain physically meaningful energies and forces
Classical orbit approach produces exact results in simple cases and mayallow for good approximations where exact solutions are nonexistent
Hope for intelligent combination of anlaytical and numerical tools for generalgeometries
Lev Kaplan (Tulane University) Quantum Vacuum Energy March 30, 2011 49 / 49